LAMIA was a child-devouring Daimon. She was a daughter of the god Poseidon, and the mother of the sea-monsters Skylla and Akheilos. Her name and family suggest she was originally imagined as a large, aggressive shark.

In one story, Lamia was a Libyan queen loved by the god Zeus. When his jealous wife Hera learned of their affair she stole away her children. Lamia went mad with grief, and tore out her own eyes. Zeus then transformed her into a monster allowing her to exact her revenge by hunting and devouring the children of others.

Lamia often appears as a bogey-monster, a night-haunting demon which preyed on children. She was sometimes pluralised into ghostly, man-devouring demon Lamiai.

The Greek word lamia means dangerous lone-shark. Such sharks were also referred to as ketea (sea-monsters). As such it is likely that she was identified with the monstrous sea-goddess Keto.
Both Lamia (Lone-Shark) and Keto (Sea-Monster) were said to have spawned the monster Skylla (the Rending One). Another child of Lamia was the boy Akheilos (the Lipless One) who was transformed into a shark by the goddess Aphrodite.

ENCYCLOPEDIA

LA′MIA (Lamia). 1. A daughter of Poseidon, became by Zeus the mother of the Sibyl Herophile. (Paus. x. 12. § 1; Plut. de Pyth. Orac. 9.) 2. A female phantom, by which children were frightened. According to tradition, she was originally a Libyan queen, of great beauty, and a daughter of Belus. She was beloved by Zeus, and Hera in her jealousy robbed her of her children. Lamia, from revenge and despair, robbed others of their children, and murdered them; and the savage cruelty in which she now indulged rendered her ugly, and her face became fearfully distorted. Zeus gave her the power of taking her eyes out of her head, and putting them in again. (Diod. xx. 41; Suidas, s. v. ; Plut. de Curios. 2; Schol. ad Aristoph. Pac. 757; Strab. i. p. 19.) Some ancients called her the mother of Scylla. (Eustath. ad Hom. p. 1714; Arist. de Mor. vii. 5.) In later times Lamiae were conceived as handsome ghostly women, who by voluptuous artifices attracted young men, in order to enjoy their fresh, youthful, and pure flesh and blood. They were thus in ancient times what the vampires are in modern legends. (Philostr. Vit. Apollon. iv. 25; Horat. de Art. Poet. 340; Isidor. Orig. viii. 11; Apulei. Met. i. p. 57.)

Source: Dictionary of Greek and Roman Biography and Mythology.

THE METAMORPHOSIS OF LAMIA

Bell, Women of Classical Mythology (sourced from Diodorus Siculus 22.41; Suidas 'Lamia'; Plutarch 'On Being a Busy-Body 2; Scholiast on Aristophanes' Peace 757; Eustathius on Homer's Odyssey 1714) (Mythology dictionary C20th) :
"Lamia was a daughter of Belus and a queen in Libya. She was very beautiful and attracted the attention of the ever-watchful and far-seeing Zeus. He had children by her, but Hera discovered their involvement and kidnapped the children. Their ultimate fate is unknown. This loss drove Lamia insane; in revenge and despair she snatched up the children of others and murdered them. The cruelty, which became obsessive, caused her appearance to change, and she became ugly with distorted features [a shark]. Perhaps in a well-intended gesture, Zeus inexplicably gave her the power to take out her eyes and then reinsert them."

Eustathius on Homer's Odyssey 1714 (Greek scholia) :
Some ancients called her the mother of Scylla.

Ptolemy Hephaestion, New History Book 6 (summary from Photius, Myriobiblon 190) (trans. Pearse) (Greek mythographer C1st to C2nd A.D.) :
"It is said that there was born also a son of Zeus and the Lamia called Akhilleus; he was of an irresistable beauty [challenged Aphrodite to a beauty contest, and was cursed with ugliness]."

LAMIA MOTHER OF SIBYLLA HEROPHILE

The Libyan Sibylla Herophile, who gave prophecies at Delphoi, was said to be a daughter of Zeus and Lamia (the Shark). This is not particularly unusual, since the oracle was frequently connected with sea-gods and creatures, especially Poseidon and the dolphin.

Pausanias, Description of Greece 10. 12. 2 (trans. Jones) (Greek travelogue C2nd A.D.) :
"There is a rock rising up above the ground [at Delphoi]. On it, say the Delphians, there stood and chanted the oracles a woman, by name Herophile and surnamed Sibylla. The former Sibylla I find was as ancient as any; the Greeks say that she was a daughter of Zeus by Lamia, daughter of Poseidon, that she was the first woman to chant oracles, and that the name Sibylla was given her by the Libyans."

Suidas s.v. Sibylla (trans. Suda On Line) (Byzantine Greek Lexicon C10th A.D.) :
"Sibylla: [The daughter] of Apollon and Lamia, though according to some of [there were various other Sibylla] . . . An Erythraian, because she was born in a region of Erythrai . . . Some supposed her a Sicilian [Sibyl], others a Leucanian, others a Sardanan, others a Gergithian, others a Rhodian, others a Libyan, others a Samian."